TY  - JOUR
ID  - Cho2011
LB  - Cho2011
AU  - Cho, Ilwoo
AU  - Jorgensen, Palle E. T.
TI  - Free probability induced by electric resistance networks on energy Hilbert spaces
ST  - Free probability induced by electric resistance networks on energy Hilbert spaces
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2011
VL  - 31
IS  - 4
SP  - 549
EP  - 598
DO  - http://dx.doi.org/10.7494/OpMath.2011.31.4.549
UR  - http://dx.doi.org/10.7494/OpMath.2011.31.4.549
AB  - We show that a class of countable weighted graphs arising in the study of electric resistance networks (ERNs) are naturally associated with groupoids. Starting with a fixed ERN, it is known that there is a canonical energy form and a derived energy Hilbert space \(H_{\mathcal{E}}\). From \(H_{\mathcal{E}}\), one then studies resistance metrics and boundaries of the ERNs. But in earlier research, there does not appear to be a natural algebra of bounded operators acting on \(H_{\mathcal{E}}\). With the use of our ERN-groupoid, we show that \(H_{\mathcal{E}}\) may be derived as a representation Hilbert space of a universal representation of a groupoid algebra \(\mathfrak{A}_G\), and we display other representations. Among our applications, we identify a free structure of \(\mathfrak{A}_G\) in terms of the energy form.
KW  - directed graphs
KW  - graph groupoids
KW  - electric resistance networks
KW  - ERN-groupoids
KW  - energy Hilbert spaces
KW  - ERN-algebras
KW  - free moments
KW  - free cumulants
SN  - 1232-9274
ER  -